How do I prove the Local intersection property in the example(Economics)

Question

$T$ is said to have local intersection property if for each $x\in X$ with $T(x)\neq\emptyset$, there exists an open neighborhood $N_{x}$ of $x$ such that $\cap_{z\in N_{x}}T(z)\neq\emptyset$.

Consider $E=F=[0,2)$ and define a correspondence $T\colon E\twoheadrightarrow F$ by $T(x)=[x,2)$.

How can I show that it has a local intersection property?